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Diffraction

Diffraction. T. Ishikawa. Part 1 Kinematical Theory. Introduction. What I intend to give in this lecture: Basic Concepts of "Diffraction" Simpler Case: Kinematical Theory (Part 1) Complicated Case: Dynamical Theory (Part 2)

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Diffraction

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  1. Diffraction T. Ishikawa Part 1 Kinematical Theory JASS02

  2. Introduction • What I intend to give in this lecture: • Basic Concepts of "Diffraction" • Simpler Case: Kinematical Theory (Part 1) • Complicated Case: Dynamical Theory (Part 2) • The speaker has been worked on experimental dynamical diffraction for 25 years. Now, he is in charge of X-ray optics for SPring-8, world's largest 3rd generation synchrotron facility in Japan. JASS02

  3. X-Rays as Shorter Wavelength Electromagnetic Wave JASS02

  4. Scattering of x-rays by a point charge (Thomson Scattering) Point Charge, mass=m, charge=e r Electromagnetic Plane Wave Lorentz Force JASS02

  5. Scattering of x-rays by distributed charge (1/2) K0 : Incident Wave Vector KS : Scattered Wave Vector KS r(x): Number Density of Electron r(x) K0 x Contribution from a volume element d3x at x JASS02

  6. Scattering of x-rays by distributed charge (2/2) 3D Fourier Transform of Charge Density Scattered Intensity JASS02

  7. Electronic Charge Distribution in Crystal Crystal = 3D Regular Stacking of Molecules Translation Symmetry N c M r(x) = r(x+la+mb+nc) l, m, n: integers b a L JASS02

  8. Fourier Transform of the Electronic Charge Distribution in Crystal JASS02

  9. Charge Density in Unit Cell Rn c b a JASS02

  10. Atomic Scattering Factor, Structure Factor Atomic Scattering Factor Structure Factor JASS02

  11. Laue Function (1/3) JASS02

  12. Laue Function, L=20 Laue Function (2/3) JASS02

  13. Laue Function (3/3) • Scattering from a crystal is appreciable only when • Then, h, k, l; integer JASS02

  14. Reciprocal Lattice (1/3) Scattering from a crystal is appreciable only when K·a = 2πh K·b = 2πk K·c = 2πl h, k, l : integer Base Vectors of Reciprocal Lattice JASS02

  15. c c* b b* a a* Reciprocal Lattice (2/3) JASS02

  16. Reciprocal Lattice (3/3) • Reciprocal Lattice Vector h, k, l; integer When the scattering vector, K=Ks-Ko, corresponds to a reciprocal lattice vector, strong diffraction may be observed (necessary condition, but not a sufficient condition). JASS02

  17. Ewald Sphere Reciprocal Lattice g Ks 2q O Ko r = 2p/l Bragg Condition of Diffraction JASS02

  18. Forbidden Reflection (1/2) K=g is a necessary condition for observing diffraction, but not a sufficient condition... If F(K)=0, then Icrystal = 0 even when K=g. Example 1, Body Center Cubic Lattice (bcc) c b a JASS02

  19. c b a Forbidden Reflection (2/2) Example 2, Face Center Cubic Lattice (fcc) JASS02

  20. Special Topics What we can measure in diffraction/scattering experiment= Intensity All phase information is lost! Non-Crystalline Charge Distribution: If is obtained, we can calculate r(x) by Fourier inversion. JASS02

  21. Iterative Phase Retrieval(Jianwei Miao & David Sayre) X-ray intensity data: Phase Information is Lost! Scattered pattern in Far Field with Coherent Illumination, Phase can be retrieved. Phase Retrieval → Iterative Algorism developed byGerchberg & Saxon, followed by the improvement by Fienup (Opt. Lett. 3 (1978) 27.) Real Space Image Scattered Intensity Phase Retrieval JASS02

  22. Reconstruction of Complex Real Space Images Real Space Phase Retrieval Scattered Intensity JASS02

  23. Original Image Reconstructed Image After 5000 iteration JASS02

  24. SEM image of Ni pattern on SiN Coherent Scattering Pattern 3D Reconstructed Image (~50 nm resolution) 2D Reconstructed Image (<10 nm resolution) 3D Diffraction Microscopy Miao et al.PRL (2002) Two Layer Ni Pattern JASS02

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